Factor score matrix

Factor scores for each of the n objects with respect to the r computed factors are then defined in the usual way by means of the nxr orthogonal factor score matrix S ... 

Using D as input we apply principal coordinates analysis (PCoA) which we discussed in the previous section. This produces the nxn factor score matrix S. The next step is to define a variable point along they th coordinate axis, by means of the coefficient kj and to compute its distance d kj) from all n row-points ... 

The goal of Q-mode FA is to determine the absolute abundance of the dominant components (i.e., physical or chemical properties) for environmental contaminants. It provides a description of the multivariate data set in terms of a few end members (associations or factors, usually orthogonal) that account for the variance within the data set. A factor score represents the importance of each variable in each end member. The set of scores for all factors makes up the factor score matrix. The importance of each variable in each end member is represented by a factor score, which is a unit vector in n (number of variables) dimensional space, with each element having a value between -1 and 1 and the... 

Other strong advantages of PCR over other methods of calibration are that the spectra of the analytes have not to be known, the number of compounds contributing to the signal have not to be known on the beforehand, and the kind and concentration of the interferents should not be known. If interferents are present, e.g. NI, then the principal components analysis of the matrix, D, will reveal that there are NC = NA -I- NI significant eigenvectors. As a consequence the dimension of the factor score matrix A becomes (NS x NC). Although there are NC components present in the samples, one can suffice to relate the concentrations of the NA analytes to the factor score matrix by C = A B and therefore, it is not necessary to know the concentrations of the interferents. 

The Q-mode varimax factor score matrix of the trace element data is shown in Table 6.3. The first factor has high positive scores for Br, As, Zn, V,... 

Q-Mode Varimax Factor Score Matrix of Trace Element Data for 88 Alberta Crude Oils... 

Squaring individual elements of the factor score matrix yielded the sum for a particular factor equal to 1.00. The proportion which an individual MM... 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

The theory of the non-linear PCA biplot has been developed by Gower [49] and can be described as follows. We first assume that a column-centered measurement table X is decomposed by means of classical (or linear) PCA into a matrix of factor scores S and a matrix of factor loadings L ... 

The principle of FA and PCA consists in an orthogonal decomposition of the original n x m data matrix X into a product of two matrixes, F (nxk matrix of factor scores, common factors) and L (kxm matrix of factor loadings)... 

X - data matrix A - factor loadings F - factor scores s - number of factors (s = m)... 

A linear combination of different factors in the matrix A with factor scores in the matrix F can reproduce the data matrix X. These factors are new synthetic variables and represent a certain quantity of features of the data set. They explain the total variance of all features in a descending order and are themselves noncorrelated. It is, therefore, possible to reduce the dimension m of the data set with a minimum loss of information expressed by the matrix of residuals E. 

If one wishes to represent the objects in the space of the factors, one has to calculate the matrix of factor scores F. The procedure is a multiple regression between the original values and the factors and is also called estimation according to BARTLETT [JAHN and YAHLE, 1970]. The graphical representation of the objects may be used to detect groups of related objects or to identify objects which are strongly related to one or more of the factors. 

Let us now come to the graphical aspects. The graphical display of the objects as factor scores is now possible after multiplying the data matrix X with each of the vectors a (factor loadings). 

The hypothesis of causes, by interpretation of the factor loading matrix, can be proved by computation of the factor scores and their graphical representation. 

Equation 11.20 describes the factorization of the experimental data matrix into two factor matrices, the loadings matrix VT and the augmented scores matrix Uau . The loadings matrix V1 identifies the nature and composition of the N main contamination sources defined by means of their chemical composition (SVOC concentrations)... 

It is seen that any (n x k) matrix can be factorized into a (n x k) score matrix T and an orthogonal eigenvector matrix. The elements of column vectors t in T are the score values of the compounds along the component vector p . Actually, the score vectors are eigenvectors to XX. ... 

Eigenvectors and eigenvalues are the products of calculation at the beginning. They characterise the property of the square matrix (correlation or covariance) derived from the initial data matrix, and they allow calculation of the factor scores F and factor loadings L, respectively. 

Principle components analysis (PCA), a form of factor analysis (FA), is one of the most common unsupervised methods used in the analysis of NMR data. Also known as Eigenanalysis or principal factor analysis (PEA), this method involves the transformation of data matrix D into an orthogonal basis set which describes the variance within the data set. The data matrix D can be described as the product of a scores matrix T, and a loading matrix P,... 

There are a variety of methods used to obtain the loading and scores matrix in Eq. (15). Perhaps, the most common methods employed are non-linear iterative partial least squares (NIPALS), and the singular value decomposition (SVD). Being an iterative method, NIPALS allows the user to calculate a minimum number of factors, whereas the SVD is more accurate and robust, but in most implementations provides all the factors, thus can be slow with large data sets. During SVD the data matrix can be expressed as... 

Principle components regression (PCR) is one of the supervised methods commonly employed to analyze NMR data. This method is typically used for developing a quantitative model. In simple terms, PCR can be thought of as PCA followed by a regression step. In PCR, the scores matrix (T) obtained in PCA (Section 3.1) is related to an external variable in a least squares sense. Recall that the data matrix can be reconstructed or estimated using a limited number of factors (/ffact), such that only the fc = Mfaet PCA loadings (l fc) are required to describe the data matrix. Eq. (15) can be reconstructed as... 

Key Words Motif discovery sequence motif sequence pattern protein domain multiple alignment position-specific scoring matrix PSSM position-specific weight matrix PWM transcription factor-binding site transcription factor promoter protein features. 

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